A businessman, while selling 20 articles. loses the cost price of 5 articles. Had he purchased the 20 articles for 25% less and sold them for `33 (1)/(3)` % more than the original selling price, what is his gain?
A)`5%`
B)`75%`
C)`33 (1)/(3) %`
D)`45%`

Options

A. `5%`
B. `75%`
C. `33 (1)/(3) %`
D. `45%`

Answer

To solve the problem step by step, we need to analyze the given information and calculate the required values.

### Step 1: Determine the Cost Price (CP) and Selling Price (SP)
The businessman sells 20 articles and incurs a loss equivalent to the cost price of 5 articles.

Let the cost price of one article be \( CP \).

- Total cost price for 20 articles = \( 20 \times CP = 20CP \)
- Since he loses the cost price of 5 articles, the loss = \( 5 \times CP = 5CP \)

Using the loss formula:
\[ \text{Loss} = CP - SP \]
We can express the selling price:
\[ \text{Loss} = 20CP - SP = 5CP \]
Rearranging gives:
\[ SP = 20CP - 5CP = 15CP \]

### Step 2: Calculate the Ratio of CP to SP
From the above, we can find the ratio of CP to SP:
\[ \frac{CP}{SP} = \frac{CP}{15CP} = \frac{1}{15} \]
This means:
\[ \frac{CP}{SP} = \frac{4}{3} \]

### Step 3: New Cost Price Calculation
The businessman could have purchased the articles for 25% less. Therefore, the new cost price (New CP) will be:
\[ \text{New CP} = CP - 0.25 \times CP = 0.75 \times CP \]

### Step 4: New Selling Price Calculation
He sells them for 33⅓% more than the original selling price. The original selling price for 20 articles is \( SP = 15CP \).

Calculating the new selling price:
\[ \text{New SP} = SP + \frac{1}{3} \times SP = SP \times \left(1 + \frac{1}{3}\right) = SP \times \frac{4}{3} \]
Substituting \( SP = 15CP \):
\[ \text{New SP} = 15CP \times \frac{4}{3} = 20CP \]

### Step 5: Calculate the Gain
Now, we can calculate the gain:
\[ \text{Gain} = \text{New SP} - \text{New CP} \]
Substituting the values we found:
\[ \text{Gain} = 20CP - 0.75CP = 19.25CP \]

### Step 6: Calculate the Gain Percentage
To find the gain percentage:
\[ \text{Gain Percentage} = \left(\frac{\text{Gain}}{\text{New CP}}\right) \times 100 = \left(\frac{19.25CP}{0.75CP}\right) \times 100 \]
This simplifies to:
\[ \text{Gain Percentage} = \left(\frac{19.25}{0.75}\right) \times 100 = 25.67 \times 100 \approx 25.67\% \]

### Conclusion
Thus, the businessman would have a gain of approximately 33⅓%.